Feller process and semi-group resolvent.

(Feller process) Let
be a Markov process in
with a homogeneous transition function
.
The
is called "Feller process" if the associated backward propagator
has the following
property:

Definition

(Resolvent of Feller process) For a
Feller process
in
we define the "resolvent"
:

Proposition

(Properties of Feller resolvent
1)

1.
,
,
,
.

2.
.

3. The range
does not depend on
.

Proof

Let
.
We verify (2) as
follows:
The last expression is symmetrical with respect to
and
and permits the change of integration order (see the proposition
(
Fubini theorem
)).

To prove (1) we
calculate
We change the order of
integration:
We make a change of variables
in the internal
integral:
We use the property
and linearity of
:

(3) is the consequence of (1).

Proposition

(Feller process property 1) Let
be a stochastic process in
with a homogeneous propagator
.
is a Feller process iff

1.
.

2.
.

Proof

For
we
calculate
We make the change
in the first
integral.

It follows from (1) that
is a linear operator in
and we conclude from the definition of
that

Hence, for
Therefore
thus
In view of the proposition
(
Properties of Feller resolvent
1
)-3, it remains to prove that
is dense in
.
We
evaluate
for large
and aim to show that the quantity is small. We make a change
in the integral,
,
The quantity
is positive and tends to 0 when
tends to
.
Hence, using
(2),
In
addition
Therefore, by the proposition
(
Dominated convergence theorem
),
for any finite measure
Hence, if the measure
vanishes
on
then it vanishes on
.
Thus,
is dense in
.

Proposition

(Properties of Feller resolvent
2) Let
be a resolvent of a Feller process. We
have